Last author copy
Submitted to Journal of Guidance, Control and Dynamics on April 1st 2015, Accepted on July 29th 2015
Collision probability due to space debris clouds
through a continuum approach
Francesca Letizia1 , Camilla Colombo2 , and Hugh G. Lewis3
University of Southampton, Southampton, United Kingdom, SO17 1BJ
As the debris population increases, the probability of collisions in space grows. Due
to the high level of released energy, even collisions with small objects may produce
thousands of fragments. Propagating the trajectories of all the objects produced by
a breakup could be computationally expensive. Therefore, in this work debris clouds
are modeled as a uid, whose spatial density varies with time under the eect of atmospheric drag. By introducing some simplifying assumptions, such as an exponential
model of the atmosphere, an analytical expression for the cloud density evolution in
time is derived. The proposed approach enables the analysis of many potential fragmentation scenarios that would be time-limited with current numerical methods that
rely on the integration of all the fragments' trajectories. In particular, the proposed
analytical method is applied to evaluate the consequences of some recent breakups on a
list of target objects. In addition, collision scenarios with dierent initial conditions are
simulated to identify which parameters have the largest eect on the resulting collision
probability. Finally, the proposed model is used to study the mutual inuence among
a set of high risk targets, analyzing how a fragmentation starting from one spacecraft
1
2
3
PhD Candidate, Astronautics Research Group, [email protected]
Lecturer, PhD, Marie Curie Research fellow at Politecnico di Milano, Italy, AIAA member, [email protected]
Senior lecturer, PhD, Astronautics Research Group, [email protected]
1
aects the collision probability of the others.
2
Nomenclature
AT = target area [m2 ]
a
= semi-major axis [km]
e
= eccentricity
err = relative error
F
= fragments' ux [km−2 s− 1]
f
= vector eld
G
= characteristic line of the system
H
= scale height for the atmospheric model [km]
h
= altitude [km]
i
= inclination [rad] or [deg]
J2
= second zonal harmonic coecient of Earth's gravitational potential, 1.082 62 × 10−3
Lc
= fragment characteristic length [m]
M
= mean anomaly [rad] or [deg]
ME = reference mass for collision [kg]
mp = projectile mass [kg]
N
= number of collisions
n
= fragment density [-/km]
RE = Earth's equatorial radius, 6.378 16 × 103 km
RH = geocentric radius of fragmentation [km]
pc
= cumulative collision probability
r
= geocentric radius [km]
t
= time [s]
V
= volume [km3 ]
v
= velocity [m/s]
vcoll = collision velocity [km/s]
vr
= radial velocity [km/s]
vθ
= transversal velocity [km/s]
w
= width of the altitude bins [km]
3
µ = gravitational constant, 3.986 00 × 105 km3 s−2
ν = true anomaly [rad] or [deg]
Ω = argument of the ascending node [rad] or [deg]
ω = argument of the periapsis [rad] or [deg]
ρ = atmospheric density [kg/m3 ]
I. Introduction
Past space missions left millions of non-operative objects in orbit and also current missions,
despite mitigation measures, continue to increase the number of debris objects because, quoting
Chobotov [1], `space debris is a self-perpetuating issue as any new space mission generates new
objects'. Currently, the focus is mostly on the largest objects of the debris population, which are
the 22 000 objects larger than 10 cm that are constantly tracked from the Earth to avoid collisions
with operational spacecraft [2, 3].
Objects smaller than 10 cm cannot be tracked with current radar technologies and, as a result,
the contribution of small fragments to the collision probability is often neglected. Objects larger
than 10 cm have also been the main scope for the evolutionary studies on the space debris population,
which analyze the long term response to the variation of some parameters such as launch frequency,
percentage of compliance with regulations, and implementation of active removal missions. However,
White and Lewis [4] showed that the eect of remediation measures is not the same for the population
of objects larger than 10 cm compared to the population between 5 and 10 cm. The latter may still
increase even when the former is expected to decrease. In other words, focusing only on the large
fragments may lead to an underestimation of the collision risk. In fact, also small fragments can pose
a relevant hazard to spacecraft. In particular, objects larger than 1 mm are yet able to interfere with
operational spacecraft causing anomalies and objects larger than 1 cm can even destroy a satellite in
case of collision [5]. Recently, McKnight et al. [6] highlighted how the so-called lethal non-trackable
objects may become the leading factor in the decrease of ight safety.
When aiming to get a global picture of space debris down to 1 mm, models currently employed
to study the debris evolution cannot be simply extended to consider also small fragments. In fact,
4
the number of objects larger than 1 cm in LEO is around 500 000 and more than one hundred
million objects larger than 1 mm are thought to be in orbit around the Earth [3]. These numbers
are too large to consider a piece-by-piece analysis of the debris population feasible as the resulting
computational time would be prohibitive. For this reason, some existing models, such as Rossi et
al. [7], sample the small fragments and dene some representative objects, which are the only ones
to be propagated. Then, the representative objects need to be re-converted into a distribution of
small fragments or in a value of fragment density to compute the resulting collision probability.
This work discusses the applicability of a novel alternative method, where the small fragments
are modeled in terms of their spatial density. This approach presents two main advantages. Firstly,
the proposed method,
Cielo (debris Cloud Evolution in Low Earth Orbits), operates directly on
the spatial density, which can be used to compute the contribution to the collision probability due to
small fragments. Secondly, the formulation in terms of spatial density admits an analytical solution
for the evolution of the fragment density in the Low Earth Orbit (LEO) region where the eect of
drag is dominant.
The method will be briey outlined in Section II, while Section III-IV present the most recent
improvements in the proposed approach. Finally, Section V discusses some possible applications of
the method, such as the study of the consequences of some real breakups on a set of targets and the
analysis of the worst conditions for a fragmentation considering the resulting collision probability
for the target set.
II. Method overview
An analytical approach to the study of space debris population in LEO was proposed by
McInnes [8]. His approach is based on the application of the continuity equation to obtain an
explicit expression for the global debris spatial density in time. In this work, instead, this approach
is used to model a single fragmentation event and assess its consequences in terms of the resulting
fragment spatial density and the change in the collision probability for other spacecraft.
For this purpose, McInnes' analytical propagation [8] is included as one of the building blocks
(Fig. 1) required to model a fragmentation event from its beginning (the breakup) to the long term
5
Band
formation
Energy
Breakup
model
Fragments
Numerical
Cloud
propagation
Density
Target
function
trajectory
Position
n0 (r)
tting
Analytical
n(r, t)
propagation
Collision
probability
Fig. 1 Cielo building blocks.
evolution. The proposed method is here briey summarized through a short description of the rst
four blocks in Fig. 1; further details on the overall method can be found in [9]. The rest of the paper
will focus on the fth block regarding the computation of the collision probability.
The simulation of a fragmentation starts with the generation of the fragments through the
application of the NASA breakup model [10, 11], considering only fragments with size between
1 mm and 10 cm. Once the fragments are generated, their orbital parameters are numerically
propagated to model the initial phase of the cloud evolution when the Earth's oblateness is the
dominant driver [12, 13]. The numerical propagator used in this phase is
PlanODyn
[14], a
semi-analytical propagator based on the averaged variation of the orbital elements in Keplerian
elements. In particular, for this work, only the following perturbations are considered: the secular
eect of the Earth's oblateness [15], considering the J2 term only, and the eect of the atmospheric
drag, considering the average variation of the parameters along one orbit as obtained by King-Hele
[16].
The numerical propagation is stopped once the fragments are spread over 360 degrees and
form a band around the Earth. The time required for this transition can be estimated through
Ashenberg's theory [17]. From this moment, the problem can be studied with the analytical method
proposed by McInnes [8], changing the point of view from the single fragments to the whole cloud.
This requires the information on the position of all the fragments to be converted into a continuous
density function n0 (r). A detailed discussion on the functions used for this purpose is provided in
Section III.
The long term evolution of the cloud is obtained by applying the continuity equation to model
the eect of atmospheric drag [8]. Using n to indicate the fragment spatial density, the continuity
6
equation is written as
∂n(r, t)
+ ∇ • f = 0,
∂t
(1)
where f = nv is the vector eld that models drag and ∇ • f is its divergence.
Assuming that the system can be considered spherically symmetrical, it can be studied through
only one coordinate, the geocentric radius r. Therefore, f has only one component, fr = vr n(r, t),
with vr is the radial velocity due to drag. Introducing the hypothesis of circular orbits for the
fragments and considering an exponential model for the atmosphere, an explicit expression for the
fragment spatial density is found
n(r, t) = n0 (ri )
ri2 vr (ri )
r2 vr (r)
(2)
where ri is the function
ri = g(r, t) = H log G(r, t) + RH
(3)
that is derived from inverting the expression of the characteristics G(r, t) at the initial time t = 0.
Further information on the mathematical details can be found in [9]. Through Eq. 2, the value of the
density at a certain altitude and at a certain time instant is immediately known and it can be used
to compute the collision probability for a spacecraft crossing the cloud as explained in Section IV.
It should be observed that the hypothesis of circular orbits for the fragments limits the applicability of the method. Firstly, it can be applied only to model fragmentations starting from circular
orbits, where the majority of the fragments have an eccentricity lower than 0.05 both in the case
of explosions and of collision. Circular orbits are, in any case, where the vast majority of cataloged
objects can be found [2] and where historically most of the fragmentations have started [18]. In addition, the hypothesis of circular orbits limits the altitudes where the method is applicable because
at low altitude (< 800 km) eccentricity has a large inuence on the accuracy of the propagation [9].
An extension of the method able to deal also with the distribution in eccentricity is currently under
development [19]. Despite this constraint, the analytical method can be still applied to study the
regions in LEO with the highest debris density, which are around and above 800 km [2]. At these
altitudes also the eect of solar radiation pressure should be considered as this force can reach the
7
same order of magnitude as drag for objects with large area-to-mass ratio. Future work will aim to
include also the eect of this perturbation in the analytical propagation, whereas the current work
considers the eect of atmospheric drag only and intend to validate the use of the spatial density
approach to compute the collision probability.
III. Density denition
A. Fragment spatial density
The simplest approach to dene the initial condition for the analytical propagation would be to
set it equal to the actual distribution of fragments with altitude at the time of the band formation
(TB ). However, in this way, the initial condition would depend on the moment when the band is
considered formed and on the specic run of the breakup model used to simulate the fragmentation.
In fact, the NASA breakup model contains some random parameters to describe the distribution of
the fragments. A Monte Carlo approach could be adopted to give statistical meaning to the results.
An alternative approach is adopted here. The positions of the fragments is not set directly as
initial condition, but the information on the fragments' orbital parameters (namely, the semi-major
axis a, the eccentricity e, the inclination i) is used to describe the fragment distribution in space,
whereas the other parameters (i.e., the longitude of the ascending node Ω, the argument of the
periapsis ω and the mean anomaly M ) are randomized within the cloud. In this way, the dependence
on the band formation time and on the run of the breakup model is reduced.
The conversion from the orbital parameters to the spatial density can be done by using the
expressions by Kessler [20] and Sykes [21], both derived from the work of Öpik [22]. They express
the probability of nding a particle, at a certain distance from the central body r and a certain
latitude β , knowing its orbital parameters a, e, i, and assuming that the other parameters can be
considered randomly distributed. Note that when a cloud generated by a breakup is simulated, the
energy will spread dierently among the fragments, so the mean anomaly M is the one randomly
distributed. These expressions depend only on geometry, so they have been applied to dierent
problems related to space debris [23, 24], the design of disposal trajectories [25], but also asteroids
[21] and Jupiter's outer moons [20]. Moreover, the dependence on the distance and on the latitude
8
can be described separately, which is particularly useful in the current application as the evolution
of the two parameters occurs with dierent time scales and drivers.
According to Kessler's [20] and Sykes' [21] expressions, the spatial density in a particle band
can be expressed as
S(r, β) = s(r)f (β)
(4)
where
s(r) =
f (β) =
1
1
r
2
4πra2
e2 − ar − 1
(5)
2
1
p
,
π cos2 β − cos2 i
(6)
so, if only the dependence on the distance is considered, Eq. 5 can be used to build the initial
condition n0 (r) = s(r). In this section only the expression as a function of the distance r is
analyzed, whereas the role of latitude is discussed in Section III C. Appendix A explains how to
derive the expression of the spatial density as a function of the orbital elements (Eq. 5) starting
from a set of fragments equally distributed in mean anomaly.
When modeling the cloud produced by a breakup, the dispersion of the orbital parameters
a, e among the fragments should be considered. This means that Eq. 5 cannot be applied directly
to describe the cloud density using the initial value of a, e of the orbit where the fragmentation
occurred. Instead, it should be applied to each fragment to take into account how the energy is
distributed among them; the total density is then obtained by simply summing the contribution of
each fragment
n(r) =
Nf
X
nj (r).
j=1
B. Validation of the density expression
The expression for the density was initially tested considering its accuracy in modeling the
initial density prole, which is the distribution of the fragments at the band formation. This was
done both on single runs of the NASA breakup model and on an average distribution obtained with
ten runs of the breakup model.
9
150
Average distribution
Fragment number
Analytical distribution
100
50
0
0
200
400
600
800
1000 1200 1400 1600 1800 2000
Altitude [km]
Fig. 2 Average fragment spatial distribution at the band formation over ten runs of the breakup
model and distribution obtained from Eq. 5. Altitude bin size equal to 25 km.
Fig. 2 shows the test performed on ten dierent runs of the NASA breakup model for a noncatastrophic collision with energy equal to 100 kJ, occurring on a circular equatorial orbit at 800 km.
The grey bars represent the average distribution of fragments from the numerical propagation and
the black lines the resulting proles applying Eq. 5.
The comparison is expressed in terms the number of fragments in an altitude shell of width equal
3
3
to 25 km, so the result of Eq. 5 is multiplied by the volume of the spherical shell Vshell = 43 π(r+
−r−
)
where r± = r ± ∆r with r center of the altitude bin and ∆r bin width. From Fig. 2 it is possible to
observe how Eq. 5 captures the general shape of the distribution and shows low variability among
the dierent runs. This observation is important because it conrms that the results obtained with
the proposed analytical method have a limited dependence on the specic run of the breakup model
used to model a fragmentation.
This aspect was studied more in detail evaluating also the variability of the results after the
propagation at dierent altitudes. For the validation, ten runs of the breakup model are used to
simulate a fragmentation; the resulting debris cloud is followed up to 1000 days after the band
formation, when the dierence with the numerical propagation is measured. The reference case is
set as the density prole obtained by averaging the result of the numerical propagation over the ten
10
runs of the breakup model. The results of the analytical propagation are compared to this reference
case but they are built starting from the output of a single run of the breakup model, so that the
variability of the results can be evaluated.
PlanODyn
[14] is used as numerical propagator also in this case, with the same settings
described in Section II. In particular, the eect of solar radiation pressure is not included in the
numerical propagation, even if its eect is important at altitudes larger than 800 km. Future work
will aim to validate the analytical propagation considering also this perturbation and will try to
include its eect in the continuity equation.
For both the propagation methods, numerical and analytical, the predicted number of fragments
still in orbit is computed and the relative error on this estimation errt is used as a measure of the
accuracy of the analytical method. In detail, errt is computed as
errt =
R
R
| nA (r) dr − nN (r) dr|
R
nN (r) dr
where nN (r) is the prole of the spatial density obtained with the numerical propagation and nA (r)
is the one from the analytical propagation. Another indicator used to measure the accuracy is the
relative error errp on the height of the peak in the distribution of fragments with altitude,
errp =
|max nA (r) − max nN (r)|
.
max nN (r)
The values of errt and errp for fragmentations at dierent altitudes are shown in Fig. 3, where
the error bars indicate the maximum and the minimum error among the ten density proles obtained
with the analytical propagation.
From the curve for errt it is possible to observe that the variability with the run of the breakup
model is very limited. For errp the variability is larger, but the error is in general lower than errp .
This shows that the proposed analytical model gives a reliable representation of the cloud evolution
without requiring multiple runs of the breakup model. This represents an advantage with respect
to other debris propagation methods that relies on multiple Monte Carlo runs.
11
0.5
0.4
0.4
0.3
0.3
errt
errp
0.5
0.2
0.2
0.1
0.1
0
700
800
900
1000
Altitude [km]
0
700
800
900
1000
Altitude [km]
Fig. 3 Relative error on the total fragment number (errt ) and on the density peak (errp ) of
the analytical model at dierent altitudes.
C. The dependence on latitude
As stated in Section III, the correct representation of the cloud spatial density requires considering also the distribution in latitude. However, in this work a constant distribution in latitude is
assumed, similar to what is already done by Kessler [24]. This approximation is chosen because the
purpose of this method is to study the long term (i.e., years) eect of a fragmentation, whereas the
latitude of a target spacecraft crossing the cloud evolves in a much shorter time scale (i.e., hours).
Following correctly the target latitude would require very short time step for the integration, eliminating or reducing the advantage of having a fast propagator for the fragment cloud.
However, it is important to remark that the analytical method is able to deal also with the
distribution in latitude. In fact, applying the general solution for a 2D formulation of a continuity
equation problem to the current application, the expression for the density can be written as
ñ(r, β, t) = ñ0 (ri , βi )
vr (ri , βi )vβ (ri , βi )
vr (r, β)vβ (r, β)
(7)
where ñ0 is the initial distribution, ri , βi are functions obtained by inverting the characteristic lines
at initial time t = 0, vr , vβ are respectively the expression of dr/dt and dβ/dt due to the eect
modeled by the continuity equation, i.e., drag. Therefore, in this case, where the eect of drag on
12
quasi-circular orbits is considered,
r − R p
dr
H
= −ε RH exp
dt
H
dβ
=0
dt
(8)
(9)
meaning that vr depends only on r and the distribution in latitude is not directly aected by drag.
As a result, Eq. 7 can be written as
ñ = ñ0 (ri , β)
vr (ri )
.
vr (r)
(10)
The expression for ñ0 (ri , βi ) is simply the one given by Kessler [20] and Sykes [21], so using the
expressions in Section III,
ñ0 (ri , βi ) = S(ri , β) = s(ri )f (β)
(11)
and nally
ñ(r, β, t) = f (β)
s(ri )vr (ri )
.
vr (r)
(12)
Similarly to what is done for s(r), f (β) can be built from the distribution of the fragments at the
time of band formation.
In this work, as explained before, the choice was not to follow in detail the evolution of the
target latitude: the collision probability is computed using an average value of the fragment density,
which depends only on the radial distance and not on the latitude.
The average density value can be found computing once the integral average of f (β) over one
orbit period and apply it to rescale the spatial density at any time, applying again the hypothesis
that the fragments' and the target's inclinations are not changing. The dependence of the latitude
β on the orbital parameters is expressed by
β = arcsin (sin (ω + ν) sin i)
(13)
where ω, ν, i refer to the argument of perigee, the true anomaly and the inclination of the target
spacecraft crossing the cloud. Introducing the argument of latitude u = ω + ν and writing the
expression for the case of circular orbit, the scaling factor of the spatial density can be computed as
f¯ =
1
2βmax
Z
0
2π
du
p
cos2
13
(β(u)) − cos2 (βmax )
(14)
100
10−1
∆ pc /pc
[-]
10−2
10−3
10−4
10−5
10−6
0
20
40
60
80
100
120
140
160
180
200
Time [days]
Fig. 4 Relative error on the collision probability due to the averaging in latitude.
where β(u) is given by Eq. 13. βmax is the maximum latitude covered by the band. For nonequatorial orbits βmax is put equal to the inclination where the fragmentation occurred iF if iF ≤ π/2
and equal to π − if otherwise. This follows from the band characterization proposed by McKnight
[12] and the observation that, with the current hypotheses (e.g., non-rotating atmosphere), the
fragment inclination is not aected by drag and so it is constant. For equatorial orbits, βmax is put
equal to the maximum inclination reached by the fragments because of the breakup.∗
This approach was tested by performing a simulation where the spatial density and the collision
probability (pc,f (β) ) are computed considering the dependence on the latitude and using a very
short time step, equal to ve minutes.† In this way, for each target orbit there are at least 20
integration points and the value of β can be considered representative of the time-step. This result
is compared to the simulation ran with a time step equal to one day, where only the dependence
on the geocentric distance is considered and the scaling factor from Eq. 14 is applied. The collision
probability obtained in this way is indicated with pc,f¯.
∗
†
This can be done for any inclination of the parent orbit and this approach was compared with the results obtained
setting the maximum covered latitude equal to the parent orbit inclination. The latter gives actually a distribution
closer to the observed one and is, therefore, implemented.
Observe that using this large number of points results in a remarkable increase in the RAM required to run the
simulation in a such a way that high-performance computing was employed for this validation. If the analytical
method is instead used on a normal machine, then a reasonable time-step needs to be chosen.
14
The result of the comparison is presented in Fig. 4: it shows the relative error
|pc,f¯ − pc,f (β) |
∆pc
=
pc
pc,f (β)
(15)
introduced by Eq. 14 on the collision probability. The simulation refers to a target with inclination
equal to iT = 60 degrees crossing a cloud generated from an orbit with equal inclination (iF = 60
degrees).
It is possible to observe how the dierence between the two methods is limited, with the relative
error that oscillates around a value equal to 0.004. Therefore, Eq. 14 is an eective way to model the
long term evolution of the collision probability without following the target latitude. For this reason,
this approach will be used in the following results where the spatial density is always computed as
S(r) = f¯s(r).
IV. Collision probability computation
Once the cloud density at any time is known, it is possible to evaluate its eect on the collision
probability for a spacecraft that crosses the cloud. The computation of the collision probability is
based on the average number of collisions N in an interval of time [20]. This number is then used to
obtain the cumulative collision probability for the target spacecraft through a Poisson distribution
pc (t) = 1 − exp (−N )
(16)
following the common analogy with the kinetic gas theory [12, 23]. According to this analogy, the
collisions between a target and a distribution of objects, in this case the fragments in a cloud produced by a fragmentation event, can be modeled similarly to the collisions among molecules within
an inert gas [26]. This requires that the probabilities of events at dierent times are independent
(fragment random motion) and that the probability of collision during a certain time interval is
proportional to the length of the time interval (large number of fragments). This approach has been
criticized for example by Chan [27], who observes how the spatial density of the fragments and the
mean free path of the target have a very dierent ratio in the case of gases and in the one of space
debris. Jenkin [26] also criticizes this approach if used in the rst phases of the cloud evolution,
when the fragments' trajectories are highly correlated. In our application, the analogy is applied at
15
a later stage of the cloud evolution, when the motion of the fragments has already been randomized.
Moreover, other approaches to the computation of collision probability, such as the one proposed
by Chan [27], should be feasible if they are based on a dependence of the number of collisions on
the fragment spatial density.
In the traditional approach, the average number of collisions N in a given interval of time
∆t = t − t0 can be written as
(17)
N = F σ∆t
where F is the ux of particles and σ represents the collisional cross-sectional area [20]. This last
parameter is usually dened considering the dimensions of both the colliding objects [20], but here
only the target spacecraft area AT is considered because the fragments are much smaller than it, so
σ ≈ AT .
The ux F is equal to
(18)
F = n(r, t)v
where n(r, t) is the value of the spatial density obtained with the
Cielo
method explained in
Section II and applying the scaling factor due to the distribution in latitude. v is the average
relative velocity between the targets and the fragments.
To keep the formulation simple and dependent only on the radial distance, a set of hypotheses
is introduced to obtain the expression of v .If a single fragment is considered, v can be obtained from
the rule of cosines
v 2 = vT2 + vF2 − 2vT vF cos φ
(19)
where vT and vF are respectively the orbital velocities of the target and of the fragment with
respect to the central body; φ is the angle between the two vectors vT and vF . vT is known from
the propagation of the target trajectory; vF is a piece of information that is lost with the analytical
propagation. However, the propagation of the fragment cloud is done under the hypothesis of
quasi-circular orbits, so
r
vF ≈ vcirc =
16
µ
.
r
(20)
Fig. 5 Generic spherical triangle.
The angle φ can be related to the geometry of the two orbits too. In fact, the intersection between
two circular orbits with the same radius can be represented by the spherical triangle in Fig. 5
where B is the ascending node of the target orbit and C the ascending node of the fragment orbit.
Therefore,
B = iT
C = π − iF ;
also a = ∆Ω, so the spherical triangle can be solved with the law of cosines to nd the angle A
cos A = sin B sin C cos a − cos B cos C
= sin (iT ) sin (iF ) cos (∆Ω) + cos (iT ) cos (iF )
(21)
(22)
Eq. 22 can be used to provide a unique value of φ for a given conguration of target and fragments
in terms of their inclinations. In fact, given that Ω is uniformly distributed among the fragments,
the average relative velocity ∆v̄ , can be found computing the integral mean of the function
∆v =
q
vT2 + vF2 − 2vT vF [sin (iT ) sin (iF ) cos (∆Ω) + cos (iT ) cos (iF )]
(23)
for ∆Ω from 0 to 2π . By putting
χ = vT2 + vF2 − 2vT vF cos (iT ) cos (iF )
η = 2vT vF sin (iT ) sin (iF )
the average value of the relative speed can be written as
2√
2η
∆v̄ =
χ+ηE
,
π
χ+η
where E[x] is the complete elliptic integral of the second kind.
17
(24)
0
180
160
30
−1
100
[deg]
80
∆i
iT
[deg]
120
−2
0
−3
log10 err
140
60
40
−4
−30
20
0
0 20 40 60 80 100 120 140 160 180
iF [deg]
700
800
hT
900
−5
[km]
Fig. 6 Relative error in the estimation of the relative velocity between target and fragments
for several congurations.
The approximation for ∆v was validated for dierent geometries of the target and fragment
orbits. The results of the validation are shown in Fig. 6, where the metric used to measure the
method accuracy is
errrel =
R
|∆vA − ∆vN | dt
R
,
∆vN dt
(25)
where ∆vN is the estimation of the velocity obtained using a numerical procedure that computes
the distance and the relative velocity between the target and each fragment; ∆vA is the analytical
estimation obtained from Eq. 24. Basically, Eq. 25 measures if the analytical approximation is able
to capture the average value of the relative velocity, which is considered to be the most relevant
parameter in a long-term study of the collision probability.
The plot on the left in Fig. 6 refers to dierent combinations of inclinations for the target
(iT ) and the fragments (iF ), while their initial altitude is the same and equal to 800 km. As one
could expect, Eq. 24 does not work for equatorial orbits, where ∆Ω is not dened; for those cases
errrel ≈ 0.3 whereas it is lower than 0.08 for all the other cases. The estimation of ∆v in the
cases with equatorial orbits could be improved if information on the distribution of the fragments
in eccentricity were available. The extension of the method towards this direction is in progress, so
future work aims to fully develop the method to consider the eect of eccentricity.
The plot on the right in Fig. 6 shows instead the results for dierent choices of the orbit
18
inclinations, with ∆i = iF − iT , and of the target altitude, with the fragmentation starting again
at 800 km. Also in this case the error is generally low, but it tends to increase with the altitude
dierence for orbits with the same inclination. In addition, it was veried that errrel is lower than
10% for the cases discussed in Section V.
Here we would like to highlight that the expression in Eq. 24, even with its limits in terms
of applicability, appears as an important improvement of the method compared to its previous
formulation [28], where it was assumed that the impact angle between the target and the fragments
is always equal to 90 degrees. This hypothesis was conservative and it largely overestimated the
relative velocity ∆v , whereas the expression in Eq. 24 provides a more realistic estimation of ∆v .
V. Collision scenarios
Thanks to its limited computational time and its good accuracy, the proposed method Cielo
can be applied to study the collision probability due to small fragments in many dierent scenarios.
The use of
Cielo is proposed for following three applications.
First, it can study the eect of a
breakup on dierent target spacecraft. Second, it can be used to build, for each target spacecraft or
for a whole set of targets, a map of collision probability by varying the inclination and the altitude
of the simulated breakup. Third, the analytical method can be applied to generate a matrix of
inuence among a selected set of targets.
The targets used for the simulations are listed in Tab. 1: they were extracted from a list prepared
by IFAC-CNR, ISTI-CNR and University of Southampton for a study sponsored by the European
Space Agency [29]. The objects in Tab. 1 are the ten spacecraft with the largest collision probability
and they are sorted by their semi-major axis. Note that the list of target spacecraft can be selected
depending on the desired application.
A. Single event simulation
The rst application of the method is the evaluation of the consequences of a breakup on the
target list in Tab. 1, considering the collision probability associated with fragments larger than
1 mm.
For this application, two recent small breakups are considered [30], whose parameters are re19
Table 1 List of target spacecraft [29] for the collision probability analysis.
ID Target
hp
[km]
ha
[km]
i
[deg]
Mass [kg]
Size [m]
1
816.0959
818.9741
98.73
4090
6.91
2
818.5311
832.9389
98.83
2490
5.17
3
804.0385
858.8315
98.83
1000
4.46
4
822.4681
865.8019
70.90
3220
4.49
5
946.4051
986.0649
82.91
1420
4.06
6
934.9528
998.1172
82.95
1420
4.06
7
964.0951
990.5749
82.95
1420
4.06
8
960.1156
1005.754
82.93
1420
4.06
9
968.9735
999.8965
82.94
1420
4.06
1099.8350
1099.8350
63.00
1000
2.41
10
ported in Tab. 2. The value in the last column is an estimation of the parameter M used in the
NASA breakup model as a measure of the energy of the breakup [10]. For non-catastrophic collisions
it is dened as the product between the mass of the smaller object mp and the square of the collision
velocity vcoll [11]
2
ME [kg] = mp [kg]vcoll
[km/s]/1[km/s].
From this parameter the fragment size distribution for a collision can be described through the
expression
N (Lc ) = 0.1(ME )0.75 L−1.71
c
(26)
where Lc is the fragment characteristic length and N (Lc ) is the number of fragments of size equal
or larger than Lc . The parameter M is here estimated considering that for the two breakups the
number of fragments added to the debris population catalog is known (respectively 35 objects for
Cosmos 1867 and 9 objects for Cosmos 2428)‡ . Therefore, assuming that the tracked fragments are
larger than 5 cm, the value of M is obtained inverting Eq. 26 and then the number of fragments NF
‡
Values updated from https://www.space-track.org/
20
in the desired size range (1 mm-10 cm) is obtained applying again Eq. 26 with the computed value
of M . Observe that the total number of fragments shown in Tab. 2 is very high and there is not
a general consensus on the reliability of the NASA breakup model in the studied size range. Some
modications of the model are available in literature [31], but the original NASA model is used
because the purpose of this work was not to develop a new breakup model, but rather to show the
possible applications and the advantages of using a formulation based only on the spatial density.
The implementation of the NASA model used in this work was validated with the comparison to
the available data on other implementations [32].
Table 2 Parameters of two recent small breakups [30].
Spacecraft
hp
[km]
ha
i0
[km]
[deg]
NLc >5 cm
M
[kg]
NF
COSMOS 1867
775
800
65
35
2.665
28138
COSMOS 2428
845
860
71
9
0.436
7235
The eect of the breakups on the target in the list is shown in Fig. 7, which shows the cumulative
collision probability caused by fragments larger than 1 mm from the time of band formation up
to ve years afterwards. The study of a single case in Fig. 7 for the COSMOS 1867 event was
obtained with an average computational time of 9.45 minutes on a cluster with 4 processors; the
average computational time is instead equal to 7 minutes for the COSMOS 2428 cases. Most of the
propagation time is required for the propagation of the fragments from the breakup to the band, so
the computational eort is required only once to generate the fragment cloud, which can be saved
and superimposed on each target spacecraft trajectory.
For the rst breakup (COSMOS 1867), the resulting collision probability pc is shown in Fig. 7(a):
it is possible to observe that the rst four spacecraft are the most aected by the fragmentation.
This is explained by two factors: rstly, SC1 and SC2 have the largest cross-sectional area, and
secondly the rst four spacecraft are both at the lowest altitudes and the shortest radial distance
from the fragmentation location. In particular, it is possible to observe an exponential relationship
between the spacecraft altitudes and the nal values of the collision probability after ve years.
A similar behavior can be observed also for the second breakup (COSMOS 2428) in Fig. 7(b),
21
×10−3
6
×10−3
6
SC1
SC2
SC3
SC4
4
SC5
SC6
pc
pc
[-]
[-]
4
2
SC7
2
SC8
SC9
SC10
0
0
500
1000
1500
0
2000
Time [days]
0
500
1000
1500
2000
Time [days]
(a) COSMOS 1867
(b) COSMOS 2428
Fig. 7 Resulting collision probability from the two breakups on the targets in Tab. 1 including
fragments down to 1 mm.
which shows the eect of inclination. In fact, the inclination of COSMOS 2428 is very similar to
the one of SC4, which, in this case, has a slightly higher collision probability than SC2 even if the
latter has a larger cross-sectional area.
B. Maps of collision probability
The collision risk for a spacecraft can be studied also from a dierent point of view: instead of
focusing on a single breakup, here the location of the breakup is changed to highlight the eect of the
breakup conditions on the collision probability. In particular, here the altitude and the inclination
of the fragmentation are changed and this allows dening the most dangerous regions for a collision
to occur for all the targets in Tab. 1. Other parameters (e.g., time, fragmentation energy) may be
considered with the same approach.
Fig. 8 shows, for example, the study done for the spacecraft SC4 for fragmentations of 100 kJ,
including all the fragments down to 1 mm. The peak in the collision probability is slightly above the
altitude of the spacecraft semi-major axis (aSC4 = RE +844 km) and for inclinations iF such that
sin(iF ) = sin(iSC4 ). Under this condition, the spacecraft will spend a part of its orbit at latitudes
where the cloud density is maximum. The collision probability is high also for inclinations where
sin(iF ) > sin(iSC4 ) because in these cases the spacecraft is always inside the band formed by the
22
Cumulative collision probability after 5 years
×10−4
100
6
80
pc
Inclination of fragmentation [deg]
120
60
4
40
20
2
0
800
820
840
860
880
900
920
940
960
980 1000
Altitude of fragmentation [km]
Fig. 8 Collision probability map for SC4 for fragmentations of
100 kJ,
including all the frag-
ments down to 1 mm.
Cumulative collision probability after 5 years
5 ×10
−3
100
4
80
pc
Inclination of fragmentation [deg]
120
60
3
40
20
2
0
800
820
840
860
880
900
920
940
960
980 1000
Altitude of fragmentation [km]
Fig. 9 Total collision probability map for the targets in Tab. 1 for fragmentations of
100 kJ,
including all the fragments down to 1 mm. The markers indicate the targets.
fragments. As expected, fragmentations at higher altitudes than SC4 have a larger eect than the
ones at lower altitudes. In fact, over time drag tends to reduce the fragments altitude and the
fragments initially at altitudes higher than SC4 decay towards the target orbit.
The same analysis was performed for all the targets in Tab. 1 to obtain the map in Fig. 9,
23
where the markers indicate the targets. Here it is possible to observe two peaks in the collision
probability, corresponding to the two bands in altitude where the targets are grouped: the rst
from 817 to 844 km for SC1-4, the second from 966 to 984 km for SC5-9. Observing the rst peak
in altitude (h ≈ 840 km), it is possible to notice that the distribution of collision probability is
not symmetrical around the peak: in fact, fragmentations at higher altitudes (e.g., 880 km) have a
larger eect than the ones at lower altitudes (e.g., 800 km) as already discussed for Fig. 8. It is also
possible to observe that the collision probability is still relatively high around h = 820 km, mainly
because of the presence of SC1, whose large cross-sectional area has a large inuence on the total
collision probability.
As observed in Fig. 7(b) and Fig. 8, the fragmentations with sin(iF ) = sin(iT ) have a large
eect on the total collision probability because of the distribution of objects with latitude. Eight
out of ten spacecraft in Tab. 1 have sin(iT ) ≈ sin(82◦ ) = sin(98◦ ), so at these inclinations two
clear bands of high collision probability are present. Compared to previous results obtained with
the continuity equation approach [33], the bands are more evident because the expression in Eq. 14
takes into account the dierent time of residence of the target inside regions with high spatial
density when its inclination is similar to the one of the fragmentation. Similar to Fig. 8, the
collision probability is high also in the whole inclination band between 80 and 100 degrees, which
corresponds to fragmentations for which all the targets are always inside the fragment band.
A map such as the one in Fig. 8 is obtained with a computational time of 3.66 hours in average§
on a cluster with 4 processors. The process can be easily automatized and parallelized to study a
list of targets and obtain a global map as the one in Fig. 9. These maps may be useful to study both
operational and non-operational targets to understand under which conditions a fragmentation has
the largest eect on the spacecraft. Moreover, the global maps can highlight the most critical areas
in terms of inuence on the whole spacecraft population and can be used, for example, to identify
interesting candidates for active debris removal. Note that the targets to build the global map can
increased or some representative objects of the whole population can be chosen.
§
among the ten cases
24
Cumulative collision probability after 5 years
2 ×10−3
10
9
8
1.5
6
1
pc
ID Source
7
5
4
3
0.5
2
1
1
2
3
4
5
6
7
8
9
10
0
ID Target
Fig. 10 Inuence matrix showing the cumulative collision probability for ten studied spacecraft.
C. Inuence matrix
The inuence matrix is proposed to study the following situation: a small breakup caused by
a non-catastrophic collision with one of the spacecraft in Tab. 1, generates a fragment cloud that
can interfere with other spacecraft. Each spacecraft in Tab. 1 is treated as a potential target and
its collision probability due to the fragment cloud is computed after a certain time. This process
is repeated scrolling through the whole list of spacecraft in Tab. 1 to obtain a picture of how each
spacecraft aects the collision probability of the other ones.
Fig. 10 shows the resulting inuence matrix for the spacecraft in Tab. 1 considering a fragmentation of 100 kJ and plotting the resulting collision probability after ve years. Here it is important
to specify that, as the proposed method is able to provide an analytical expression for the density
only after the band is formed, the collision probability is computed starting from that moment.
This means that the collision probability may be underestimated for satellites such as SC5-SC9 that
have very similar orbits and that may start to interact before the band is formed.
The sum of the collision probabilities, over all the targets, due to the same source can be used
as an index of the spacecraft inuence ; similarly, the sum of the collision probabilities for one target
from all the sources can be used as an index of its vulnerability. In formulas, I(i, j) is the element
25
×10−2
1
Inuence
Vulnerability
0.8
P
pc
0.6
0.4
0.2
0
1
2
3
4
5
6
7
8
9
10
ID Source
Fig. 11 Sum of the generated collision probability for the scenarios in Fig. 10.
of the inuence matrix that expresses the cumulative collision probability of the object j due to a
fragmentation starting from the object j . For a generic object k, the two indices are obtained from
Inuence(k) =
N
tot
X
I(k, j)
j=1
Vulnerability(k) =
N
tot
X
I(i, k).
i=1
Both these values are shown in Fig. 11.
As one can expect, the inuence is very strong among satellites on similar orbits such as the
already cited group SC5-SC9 and the group SC1-SC4. SC10 has, instead, the lowest inuence because it is in an orbit with lower inclination than the other objects and, therefore, its fragmentations
aect a smaller range of latitudes.
A high vulnerability is registered for SC1, which is in a lower orbit than the other spacecraft
and which has a much larger cross-sectional area. This explains why it is aected by all the fragmentations originating from the other spacecraft. On the other hand, SC10 is the least vulnerable
target because of its high altitude (with more than 115 km of separation between its semi-major
axis and the one of the closest object) and because of its small cross-sectional area.
The computational time required to generate Fig. 10 is equal to 645 s on a PC with 8 CPUs at
3.40 GHz. The process is fully automatic and parallized, so the number of spacecraft in the list can
26
be extended to obtain a more complete picture of the mutual inuence among what are considered
the most critical objects in the debris population in LEO.
VI. Conclusion
Small debris fragments are often not included in the study of the evolution of the debris population even if they can still pose a relevant hazard to spacecraft in case of collision. The number
of small fragments is too large to follow each object separately, so a method to study them in
terms of their resulting spatial density was proposed. This requires converting the information on
the position of the fragments into a continuous density function. The approach used derives the
spatial density from the fragments' orbital parameters and not directly from their positions. It was
shown how in this way the results are less dependent on the the run of the breakup model used to
produce the fragment cloud. Once the initial density prole was dened, its evolution with time
under the eect of drag was obtained by applying the continuity equation, which allows deriving
an explicit expression for the density as a function of time and distance. The dependence of the
fragment density on the latitude was instead neglected as the focus was on the long term evolution
of the cloud, whereas the latitude of a possible target spacecraft evolves on a much shorter time
scale. The explicit expression for the density allows the method to provide a very fast estimation
of the extension of the region of space aected by the fragmentation and of the resulting collision
probability for a spacecraft in that region. For this reason, the proposed method can be applied
to simulate many collision scenarios in a short time, enabling new analysis on the contribution of
small fragments to the collision probability. In particular, here the method was applied to study
the mutual inuence among a list of spacecraft in the case they originate a fragment cloud as a
result of a small breakup. The resulting matrix of collision probability can be useful to identify
which objects, in case of fragmentations, are more likely to have a large eect on the global collision
probability and are therefore critical items in the debris population.
Appendix A
The expression of the spatial density can be obtained starting from the hypothesis of mean
anomaly M equally distributed. In this case the density nM (M ) will be constant with M and the
27
follow condition holds
Z
2π
(27)
k dM = 1.
0
The rst step is to obtain the distribution with the true anomaly ν . Starting from the denition
of M
(28)
M = E − e sin E,
with E eccentric anomaly, dM can be written as
dE
dE
dM =
− e cos E
dν
dν
dν
" √
#
√
1 − e2
e + cos ν
1 − e2
=
−e
dν
1 + e cos ν
1 + e cos ν 1 + e cos ν
(29)
(30)
3
(1 − e2 ) 2
=
dν.
(1 + e cos ν)2
(31)
Therefore,
3
nν (ν) =
(1 − e2 ) 2
(1 + e cos ν)2
(32)
which is identical to the expression by McInnes and Colombo [34], beside a constant term due to
the dierent choice in the normalization.
The following step is the translation into a distribution in r. Starting from the denition of r
the expression for dν is found.
r=
a(1 − e2 )
1 + e cos ν
⇒
dr =
a(1 − e2 )
e sin ν dν
(1 + e cos ν)2
(33)
It is convenient to express sin ν as
r
sin ν =
p
1−
=
p
1 − e2
cos2
a
er
i2
1 ha
2) − 1
(1
−
e
e2 r
r
2
e2 −
−1
a
1−
ν=
r
(34)
(35)
so that
dν =
(1 + e cos ν)2 r
q
3
(1 − e2 ) 2 a2
1
e2 −
r
a
2 dr.
−1
(36)
Substituting dν in
Z
3
(1 − e2 ) 2
k
dν
(1 + e cos ν)2
28
(37)
one obtains
Z
k
r
q
a2
1
e2 −
r
a
(38)
2 dr
−1
so the distribution nr (r) in r is
nr (r) = k
r
q
a2
1
e2 −
r
a
(39)
2 .
−1
Eq. 39 has as dimensions [1/km]; to obtain a real spatial density other two steps are required.
Firstly, the number of objects in a bin are counted and secondly, the number is divided by the
volume of the shell dened by the altitude bin.
For the rst step, a rigorous approach will require
r+δh
Z
N (r; ∆h) =
k
r
r
1
q
2 dr;
a2
e2 − ar − 1
(40)
if we consider ∆h → 0,
Z
r
N (r; ∆h) =
k
−∞
r
q
a2
(41)
Z
1
e2 −
r
a
−1
2 dr−
r+∆h
k
−∞
r
q
a2
1
r
a
e2 −
r
1
2 dr ≈ k a2 q
2 ∆h.
−1
e2 − ar − 1
Similarly the volume of the shell can be written as
V =
4
4
π[(r + ∆h)3 − r3 ] = π[3∆hr2 + O(∆h2 )] ≈ 4πr2 ∆h.
3
3
(42)
The spatial density s is nally obtained as
s(r) ≈
nr (r)∆h
k
q
=
4πr2 ∆h
4πa2 r
1
e2 −
r
a
2 ,
−1
(43)
which is identical to Kessler's [20] and Sykes' [21] expression apart from a constant (k = 1 in his
expression).
VII. Acknowledgements
Camilla Colombo acknowledges the support received by the Marie Curie grant 302270 (SpaceDebECM - Space Debris Evolution, Collision risk, and Mitigation), within the 7th European Community Framework Programme. The authors acknowledge the use of the IRIDIS High Performance
Computing Facility, and associated support services at the University of Southampton, in the completion of this work. The authors would like to thank the editor and the reviewers for the detailed
comments that contributed to improve the quality of the paper.
29
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